Those caveats are just the beginning. You generally also want the mean to be associative, you want the function they call 'g' to be invertible, and the harmonic

Those caveats are just the beginning. You generally also want the mean to be associative, you want the function they call 'g' to be invertible, and the harmonic mean and geometric mean have the simpler and more useful interpretation of taking the average of the reciprocal and logarithm respectively. All this article is doing, in a somewhat confused fashion, is trying to replace addition with another abelian (but not associative) operator. Which is fun, but ultimately not better than just assuming the arithmetic mean is going to be any help (which at least has some strong theorems that tell you it's going to converge to the expected value, usually).